How many color patterns can you create by reversing the colors of combinations of rows and columns from a checkerboard?

OK @LostInParadise – I understand each word individually, but I don’t have a friggin clue what you are asking.

Can you perhaps explain the question better for simpletons like me?

No matter the number of squares and columns, each square is in only one column and one row. So if you reverse each row and each column, you will reverse each square twice, so it will revert back to its original state.

@elbanditoroso , Take a row of a checkerboard. Suppose its colors are red,black, red, black, red, black, red, black. If you select the row then the colors will reverse to black, red, black, red, black, red, black, red.

Choose a group of rows and columns and, one at a time, reverse the colors in each chosen row and column.

@zenvelo, That is correct. That means that the number of color patterns must be less than the number of ways of selecting rows and columns. It is tempting to say that the number of color patterns is one fewer than the number of ways of choosing rows and columns, but the number is actually much less. Do you see why?

Here is a hint on how to approach this problem. For any pattern of rows and columns, each row and column of the checkerboard is chosen either zero or one time. That gives 2^16 ways of choosing rows and columns. By limiting the number of times a row or column is chosen to 0 or 1, we have made use of the fact that repeating a column or row cancels the operation. We still need to account for the fact that choosing all rows and columns also cancels everything out.

Suppose we choose the first row. After that, choose all rows and columns. We know the net effect will be the same as just choosing the first row. We also know that the the two operations both choose the first row, so the net effect can also be looked as being the same as choosing everything except the first row. Do you see where this reasoning leads?

Just to close this out, each color pattern can be created in two ways, by both a combination of rows and columns and also by the complement of that combination – all of the rows and columns that were not chosen. That means the total number is ½ (2^16) = 2^15 = 32,768. Another, easier problem, is to figure out how to choose rows and columns so as to end up with squares all of the same color.